Hit points, sometimes abbreviated as HP, is the basic unit of health in the Stamina modes of the Super Smash Bros. Melee and Brawl, as well as the unit of health for bosses and the enemies in each game's Adventure mode. When a character is hit by an opponent's attack, their hit points decrease. The character can continue as normal until their hit points are reduced to 0, when they are KO'd. The health of two recurring bosses in the Super Smash Bros. series, Master Hand and Crazy Hand, are always measured in hit points, unless otherwise altered by the Master Hand glitch or an Action Replay.
Hit points of a character decreases just like damage is raised, so if an attack did 10%, it would decrease the opponent's HP by 10 points. Once the HP reaches 0 the character (or boss) will show an animation sequence of that character fainting, during which the sequence the other characters move in slow motion. In the case of Adventure mode enemies, when their HP is reduced to 0, they'll be sent flying back until they pass a blast line or disappear.
In terms of game coding, HP in Stamina battles is equal to a negative damage - that is, an HP displaying as 123 HP is actually a damage of -123%. This explains why characters take minimal knockback in stamina matches until they are defeated (HP drops to 0, damage begins to rise above 0%). In the same way that damage is displayed as an integer but tracked as a decimal (so a damage of 14.6% is reported as 14%), HP is also rounded down - though as it's considered negative, it is rounded up, so a value of 14.6 HP would be displayed as 15 HP; this has the side effect of any HP value less than 1 (such as 0.4) being reported as 1, so it is impossible to display a 0 if the character is not defeated. On the other hand, HP for enemies and bosses (at least in Brawl) is a positive number that is decreased on taking damage.